geometric figures - singapore math · · 2014-06-0928 • geometric figures 1. ... two...

CHAPTER 2Geometric Figures

This chapter describes how elementary students are introduced to the world of geometry. Wehave seen how children learn to measure lengths and angles and to solve arithmetic problemswith measurements. At the same time, in a separate part of the curriculum, geometry begins asa subject in its own right.

Geometry is the study of relationships among the measurements – lengths, angles, areasand volumes – of figures. Already by grade 2, geometry moves beyond naming figures: theactivities direct attention to lengths and angles. In grades 2–4, children learn about paralleland perpendicular lines and solve problems involving supplementary and vertical angles. Theyalso learn to draw figures using a rulers, protractor, compass and set square. In grades 5 and 6the pieces come together and they begin solving problems involving lengths and angles withintriangles and quadrilaterals.

The elementary geometry curriculum focuses on clear reasoning and simple geometric facts.There are no proofs; instead, facts are introduced using paper-folding or symmetry arguments.Reasoning skills are developed through daily problem sets, problem sets that can be great funfor both children and teachers.

2.1 Fundamental Geometric Ideas

As children learn to measure in grades K-3 they acquire intuition about segments, angles andother objects of geometry. Sometime near the end of elementary school, intuition is replacedby precise definitions. This section describes how angles, perpendicular and parallel lines, andfigures are presented intuitively, and then in the precise but child-friendly form of a “schooldefinition”.

Angles

The “angle measurement facts” listed on page 24 have three especially useful consequences:

27

28 • Geometric Figures

1. The total measure of adjacent angles arounda point is 360!. (Abbreviation: !s at a pt.)

bºa+b+c = 360.

aº cº

2. The total measure of adjacent angles forminga straight line is 180!.(Abbreviation: !s on a line.)

bºcº

a+b+c = 180.

aº

3. The sum of adjacent angles in a right angle is 90!.(Abbreviation: !s in rt. !.)

aºbº

a+b = 90.

As you will see, these facts will be used repeatedly; they are the springboard that launchesdeductive geometry in grades 5 and 6. We have given each an abbreviation. Learning geometryis easier in classrooms where everyone consistently uses the same short, clear abbreviations.

When two lines intersect, they form four angles around the point of intersection. If we knowthe measure of any one of these, we can determine the measures of all four by recognizing pairsof supplementary angles.

EXAMPLE 1.1. In the figure, find the measures of angles b, c and d.

40º

bºcº

dº

Solution: Moving around the vertex clockwise,we see successive pairs of supplementaryangles:

40 + b = 180, so b = 140.b + c = 140 + c = 180, so c = 40.c + d = 40 + d = 180, so d = 140.

In the figure below, angles a and c are a pair of vertically opposite angles because they areverticalangles opposite each other through the vertex. For short, one says that a and c are vertical angles.

Teachers should avoid the phrase “opposite angles” because pairs of angles that might be called“opposite” occur in other, di!erent contexts (see page 48).

The figure contains two pairs of supplemen-tary angles. Angles a and b are angles on a line,so a is 180"b. Similarly, c is also 180"b. There-fore a = c, so the two vertical angles have equalmeasure.

aº

bºcº

SECTION 2.1 FUNDAMENTAL GEOMETRIC IDEAS • 29

Vertical angles have equal measure.(Abbreviation: vert. !s.)

aº

a = c.

cº

EXERCISE 1.2. Read pages 85–88 in Primary Math 5A. Which angle facts are introduced onthese pages?

Perpendicular and Parallel Lines

As we saw in Section 1.4, students are introduced to right angles in Primary Math 3B. Thenin Primary Math 4A, they learn that a right angle measures 90! and, a few pages later, they learnthe term “perpendicular”.

DEFINITION 1.3. Two segments, rays, or lines areperpendicular if the lines containing them inter-sect to form a 90! angle.

If#$AB is perpendicular to#$CD, we write#$AB % #$CD.

a small square indicates a 90º intersection

Two intersecting lines form four angles. If one of those angles is 90!, then by symmetryeach of the other angles must be 90!.

Drawing perpendicular lines is harder than recognizing them because it requires motorskills. Hands-on activities drawing perpendicular lines deepen students’ understanding andset squaredevelops skills that will be useful later. In classrooms, right angles are usually drawn with theaid of a set square (or plastic triangle), although any object with a right angle such as a pieceof cardboard can be used.

EXAMPLE 1.4. Use a set square to draw a perpendicular to the line#$AB through the point C.

C

A

B

C

A

B

C

A

B


Two lines are parallel if they lie in the same plane and do not intersect. In addition, in K-8parallel lines geometry — but not always in high school geometry — a single line is considered parallel to

itself. When lines #$AB and #$CD are parallel, we write #$AB!#$CD. In pictures, pairs of matchingarrows are used to indicate that two lines are parallel.

A

BC

DAB//CD

Two segments are parallel if they are part of parallel lines.parallel segments

As always, the word “line” means a straight line that extends indefinitely in both directions.Consequently, to make sense of the phrase “do not intersect” children must envision extendingthe segments on their paper indefinitely into space, past distant galaxies — something that is notvery concrete. Furthermore, the condition that two lines do not intersect does not suggest a wayof drawing parallel lines, or a way to determine when two segments are parallel. Consequently,a di!erent definition of parallel lines is used in elementary school.

DEFINITION 1.5 (School Definition). Two lines, segments, or rays are parallel if they lie in thesame plane and are both perpendicular to a third line.

This school definition gives a mental picture that is concrete and easily explained. Examplesof segments that have a common perpendicular are all around us, while there are few examplesof non-intersecting lines extending indefinitely into space past distant galaxies.

Railroad tracks and lined paperhave many parallel lines.

After describing parallel lines, the Primary Mathematics curriculum focuses on two activ-ities to help students develop intuition about parallel lines: a method for determining whethertwo lines are parallel and a method for constructing parallel lines using a set square. The firstmethod calls children’s attention to a fact that is treated as common knowledge in elementarymathematics: if two lines are both parallel to a third line, then they are parallel to each other.


EXAMPLE 1.6. Use a set square to determine whether two lines are parallel.

C

DA

B

01

23

45

67

89

C

DA

B

01

23

45

67

89

C

DA

Are these lines parallel? Place set-square, then ruler. Slide.

Look closely!Answer: No.

B

EXAMPLE 1.7. Use a set square and a ruler to draw a line parallel to#$AB through the point C.

CA

B

0

1

2

3

4

5

6

7

8

9

CA

B

0

1

2

3

4

5

6

7

8

9

CA

B

The Primary Math books include a marvelous set of problems involving parallel lines withinparallelograms and other figures (see Section 2.4). Then in seventh grade, students study per-pendicular and parallel lines more abstractly.

Circles

Children learn to recognize and name circles in kindergarten and first grade, but it is usuallynot until third or fourth grade that they encounter the precise definition of a circle. At this timethey learn the terms “radius” and “diameter”, and learn to draw circles with compasses. Here isa three-step teaching sequence for introducing circles:


Step 1 — Definition. Circles are defined in terms of distance, not visual shape. The followingactivity helps children understand the key idea.

02

46

810

13

57

9

P

Mark point P on a transparency. Then mark as many points as you can that are 6 cm from point P.

AAAPPPP

Put your transparency on top of your classmates’ papers.

What do you notice?

This activity shows that (i) a circle is a collection of points, and (ii) a circle is completelydetermined by its center and its radius. With these two realizations, children understand theessence of the mathematical definition

DEFINITION 1.8. Choose a point P in the plane and a distance R. The circle with center P andcirclecenterradius radius R is the set of all points in the plane that are distance R from the point P.

The word “radius” has two meanings. The radius of a circle is a distance, as in Defini-tion 1.8, while a radius is any segment with one endpoint on the center and the other endpoint

radius on the circle. (The plural of this word is “ radii”, pronounced “ray-de-eye”.) Because doublemeanings engender confusion, alert teachers always clarify whether “radius” refers to a distanceor a segment.

In a circle all radii have the same length. Teachers can reinforce this aspect of the definitionby having students use rulers to check that all radii have the same length.

center

radius

A

B

CP

D E

F

Step 2 — Drawing Circles. A compass is a tool for drawing circles. Learning to use a compasscompassis a prerequisite for later geometry.


EXERCISE 1.9. Mark a point P on your paper. Draw a circle with center P and radius 5 cm.

0 1 2 3 4 5 6 7 8 9 10

Adjust your compass to 5 cm.

Place the compass point on P. Rotate to draw a circle.

Use one hand, not two.

Lean the compass forwardand pull - don’t push - the pencil.

Don’t let the point of thecompass slip off P.

As in Definition 1.8, a circle is determined by choosing a center point and choosing a radius.These are exactly the choices students make when they use a compass to draw circles. Othermethods for drawing circles, such as tracing around a tin can, hide those choices. The simplicityof the definition gets lost!

Using a compass to draw circles is easy, but requires some practice. The quality of thecompass is a factor here. Compasses with screw adjustments work well, but the arms of low-quality compasses tend to slip, making it frustratingly di"cult to maintain a constant radiuswhile drawing circles.

Step 3 — Properties of Circles. A line and a circle can intersect in zero, one, or two points.

disjoint one intersection pointtangent circle

two intersection points

Likewise, two di!erent circles intersect in zero, one, or two points.

disjoint one intersection pointtangent line

two intersection points

EXERCISE 1.10. Use a compass and a straightedge to draw two circles that are tangent. (Hint:Draw a circle with center P and a ray with endpoint P. How can you choose a center point anda radius for a second, tangent circle?)


EXERCISE 1.11. The following third grade problem describes circles that intersect in two points,but the problem has only one solution. Why?

Gold is buried 4 cm from point B and 2 cm from point C. Use your compass to find thetreasure.

A

B

C

D

The teaching sequence continues in Primary Mathematics textbooks in sixth grade withdefinitions and applications of terms like diameter, circumference, and semicircle (see pages26–29 of Primary Math 6B). We will examine this phase of the curriculum in Chapter 8.

Homework Set 5

1. For each of the following times of day, sketch a clock faceshowing that time and find the measure (in degrees) of theangle formed by the hour hand and the minute hand. (Re-member that the hour hand moves 30! per hour.)

a) 3:00 b) 3:30 c) 10:30 d) 2:45.

2. In the figure, a = 38!. Find b, c, and d.

a=38ºb

cd

3. On page 238 of NEM1, do Problem 1 of Class Activity 2.

4. (Study the textbook!) Read the rest of Class Activity 2.These exercises have students accept three basic facts af-ter checking them in one example. These are not proofs;the purpose of the activity is to make clear what the facts

mean and help make them evident to the students (ratherthan being teacher-announced truths).

a) Copy down the three facts with their abbreviations.We will shorten the first two abbreviations to “!s ona line” and “vert. !s”.

b) Are these the same three facts introduced on pages85-86 of Primary Math 5A?

5. In Exercise 9.2 on pages 240-241 of NEM1, do problems1b and parts b), f), h) and j) of Problem 2. Write yoursolutions in the manner of Worked Examples 1 and 2 onpages 239 and 240 on NEM 1. Be sure to include reasonsin parentheses.

6. (Study the textbook!)a) Read pages 78–83 of Primary Math 4A. Does the

book define the terms perpendicular and parallel, ordoes it just show examples?

b) Now read pages 242–243 of NEM1. Does NEM1define the terms perpendicular and parallel?

SECTION 2.2 TRIANGLES • 35

c) Give a one or two sentence explanation of how Prob-lem 2 on page 80 of Primary Math 4A helps studentsmake sense of the seventh grade definition of per-pendicular.

d) Give a similar explanation of how Problem 2 on page83 helps students make sense of the seventh gradedefinition of parallel lines.

7. On page 243 of NEM1, do Problem 1 of Class Activity 3using a ruler and a set square. Then write a precise defi-nition for the term perpendicular bisector.

8. (Common Student Error) When asked to find x in the fig-ure below, Mary writes “x = 50” and explains “becausevertical angles are equal”. What is x actually? What erro-neous assumption did Mary make?

x°

160°

50°

9. (Common Student Error) In the figure, Jerry claims that“a + b = 180” stating as the reason, “!s on a line.”

aº bº

There may be several reasons why Jerry is making this er-ror. He may not understand angle measurements or how

to add angle measurements, he may not understand thatto apply this fact the angles must be , or hemay not understand that there are degrees on astraight angle.

10. Draw a line L and a point P not on L. Using a ruler andset square, follow the procedure of Example 1.7 in thissection to draw a line through P parallel to L.

11. A teacher asks her students for a precise definition of theterm circle.

a) Sarah says “A circle is a round segment with noendpoints”. Name at least two things wrong withSarah’s definition.

b) Michael says “A circle is 360!”. What two notionsis Michael confusing?

c) Write down a precise definition of the term circle.

12. Practice drawing circles with a compass until you candraw complete circles without stopping. Then on yourhomework paper draw circles of radius 2 cm, 6 cm and10 cm, using your ruler to set the compass width.

13. Use your compass to draw a circle; label your circle Cand its center O. Choose any point on the circle and labelit P. Draw another circle of radius OP with center P.

14. Use your compass to draw circle B; label its center 0.Choose any point on the circle and label it Q. Draw twomore circles of di!erent radii such that both circles inter-sect circle B only at point Q. (Hint: Draw line#$OQ first.)

2.2 Triangles

This section explains how an elementary school teacher might introduce one of the mostimportant ideas in geometry: the fact that the sum of measures of the interior angles in anytriangle equals 180!. To set the stage, we briefly discuss how children are introduced to shapesand figures in grades K-2.

Geometric Figures in the Early Grades

Pre-school, kindergarten, and first grade children learn to name, identify and draw 10 " 15types of figures in the plane. There are two distinct levels to this learning. At the lower level,students identify shapes by sight-recognition: a square is a shape that looks like a square. Shape-naming is easy because young children have an innate ability to recognize and match shapes andare dazzlingly quick at learning new vocabulary words. At the higher level, students learn a pre-cise definition and learn to check that the figures satisfy the required conditions. A “hexagon”,for example, is not simply something that looks like a hexagon. Rather, a hexagon is a figurewith six straight sides — count ‘em to be sure! The teacher’s role is to move students from thefirst level to the second.

CHAPTER 3Finding Unknown Angles

Geometry becomes more interesting when students start using geometric facts to find unknownlengths and angles. During this stage, roughly grades 5-8, students work on “unknown angleproblems”. These problems are learning bonanzas. They initiate students in the art of deduc-tive reasoning, solidify their understanding of geometry and measurement, and help introducealgebra.

You have already solved some unknown angle problems and seen how they are integratedinto the Primary Math curriculum in grades 5 and 6. This chapter examines how unknown angleproblems are used to develop geometry in grades 6 and 7.

From a teaching perspective, unknown angle problems are not just part of the geometrycurriculum, they are the curriculum in grades 5-8; everything else is secondary. In these grades,teachers and textbooks introduce facts about angles within triangles and polygons, about par-allel lines, about congruent and similar figures, and about circles. These are not simply factsto memorize: understanding emerges as students use them to solve problems. Thus teachingcenters on solving problems.

Unknown angle problems are superbly suited for this purpose. Solutions require severalsteps, each applying a known fact to the given figure. As students do these problems the ge-ometric facts spring to life; these facts become friends that can be called upon to help solveproblems. Unknown angle problems are also enormous fun!

3.1 Unknown Angle Problems

An unknown angle problem is a puzzle consisting of a figure with the measures of somesides and angles given and with one angle — the unknown angle — marked with a letter. Thestudent’s task is to find the measure of the unknown angle by applying basic geometric facts.Beginning exercises require only rudimentary facts, such as the fact that angles around a pointadd to 360!. As new geometric facts are introduced, they are added to the list of facts thatare available as tools to solve unknown angle problems. As more knowledge is integrated, theproblems become more challenging and more interesting.

55

56 • CHAPTER 3. FINDING UNKNOWN ANGLES

This section examines the role of unknown angle problems in the Primary Math and NewElementary Math textbooks for grades 5-7. It includes a list of the geometric facts learnedduring this stage and a format for presenting “Teacher’s Solutions” to unknown angle problems.You will be asked to use this format for many homework problems.

Many elementary textbooks, including the Primary Math books, introduce new conceptsusing the following specific process.

Teaching sequence for introducing geometric facts

1. Review background knowledge and introduce any new terms needed.

2. Introduce the fact by an activity (measuring, folding, or cutting-and-rearranging) thatserves to clarify what the fact says and convince students that it is true.

3. Summarize by stating the geometric fact in simple clear language.

4. Have students solve dozens of unknown angle problems:

a) simple problems using the fact alone,b) multi-step problems using the fact alone,c) multi-step problems combining the fact with previously-learned facts.

Step 3 takes only a few minutes, but it is the teacher’s most important input. In geometry,words have precise meanings; students’ success depends on knowing definitions and knowinghow to apply them. One can even argue that geometry is included in the K-12 curriculum toteach students that giving words precise meaning fosters clear thinking. This lesson is applica-ble to all subjects.

After these preliminaries, the fun begins as students solve increasingly challenging prob-lems (Step 4). As always in mathematics, the real learning occurs as students solve problems.

Geometry Facts — First List

As you have seen in homework problems, the basic facts about angles, triangles and quadri-laterals are presented in Primary Mathematics 5A and 5B. Below is a list of the facts learned atthat stage. Each has a simple abbreviation. You will be expected to be consistent in using theseabbreviations in your homework solutions.

The list of facts is built around three exercises. These questions ask you to observe howthese facts are justified at the grade 5 level (using folding, cutting, and measuring exercises) andto observe the type of problems students are asked to solve.

EXERCISE 1.1 (Angle Facts). The following three facts are introduced on pages 85–88 of Pri-mary Mathematics 5A. How are these facts justified?

SECTION 3.1 UNKNOWN ANGLE PROBLEMS • 57

Vertical angles have equal measure.(Abbreviation: vert. !s.)

bºaº

a = b.

The sum of adjacent angles on a straightline is 180!. (Abbreviation: !s on a line.)

bº

a+b = 180.

aº

The sum of adjacent angles around a point is 360!.(Abbreviation: !s at a pt.)

bºa+b+c = 360.

aº cº

EXERCISE 1.2 (Triangle Facts). The following five triangle facts are introduced on pages 57–64 in Primary Mathematics 5B. Locate the statement of each in your 5B book. What activity isused to justify the first fact? What wording is used for the fourth one?

The angle sum of any triangle is 180!.(Abbreviation: ! sum of !) a°

b°

c°

a + b + c = 180.

When one angle of a triangle is a right angle,the other two angles add up to 90!.

(Abbreviation: ! sum of rt. !.)a°

b°

a + b = 90.

The exterior angle of a triangle is equal tothe sum of the interior opposite angles.

(Abbreviation: ext. ! of !.)a° d°

b°

DCA

B

d = a + b.


Base angles of an isosceles triangle are equal.

(Abbreviation: base !s of isos. !.) a°

a = b

b°

Each interior angle of an equilateral triangle is 60!.

(Abbreviation: equilat. !.)60° 60°

60°

EXERCISE 1.3 (Quadrilateral Facts). The next section of Primary Math 5B (pages 68–71)introduces two facts about 4-sided figures. Study the folding and cutting exercises given onpage 70. How would you use these exercises in your class?

Opposite angles in a parallelogram are equal.

(Abbreviation: opp. !s!-ogram.) a°

a = b.

b°

Interior angles between two parallel sides in a trape-zoid (or a parallelogram) are supplementary.

(Abbreviation: int. !s, BC!AD .)a°

c°

d°

b°

a + b =180.c + d =180.

A

B C

D

The “Teacher’s Solution” Format for Unknown Angle Problems

Teachers are obliged to present detailed solutions to problems for the benefit of their stu-dents. The teacher’s solutions must meet a di"erent standard than the students’ solutions. Bothteachers and students are expected to get the reasoning and the answer correct. But teacher-presented solutions must also communicate the thought process as clearly as possible.

In this book, solutions that meet this high standard are called Teacher’s Solutions. You willfrequently be asked to write such Teacher’s Solutions in homework. If you are unsure how to dothis, look in the textbooks: almost every solution presented in the Primary Math books, and allof the “Worked Examples” in the New Elementary Mathematics book, are Teacher’s Solutions.

You are already familiar with one type of Teacher’s Solution — bar diagrams. Bar diagramsare extraordinarily useful for communicating ideas about arithmetic. Teachers need similardevices for communicating geometric ideas. As a start, in this chapter you will be writingTeacher’s Solutions for unknown angle problems. Here is a simple example.

SECTION 3.1 UNKNOWN ANGLE PROBLEMS • 59

EXAMPLE 1.4. The figure shows angles around a point. Find the value of x.

. ..

3x + 15 = 75

3x = 60

x = 20.

Teacher’s Solution to an Unknown Angle Problem

Answer clearly stated on the last line.

Diagram showsall needed information.

A new line for each step.

Each fact used is stated on the same line using our abbreviations.

75º 3xº

15º

vert. s.

}Arithmetic and algebra done one step at a time (no reasons needed).

This solution is short and clear, yet displays all the reasoning. It always begins with a pictureshowing all points, lines, and angles used in the solution, and it always ends with a clear answerto the question asked.

Notice what happens with the degree signs. The angles in the picture have degree signs, so75, 15 and 3x are all numbers. Thus we can drop the degree signs in the equations. This saveswork and makes the solution clearer. Degree signs are handled in the same way in the next twoexamples.

EXAMPLE 1.5. The figure shows a parallelogram. Find the value of x.

Teacher’s Solution:

x°

a°

74°

67°

A D

BC

a = 74 opp. !s of!-ogramx = a + 67 ext. ! of !ABD

" x = 74 + 67= 141.

EXAMPLE 1.6. Find the values of a and b in the following figure.

Teacher’s Solution:

bº aº

44º

66º

95º

a + 44 + 95 = 180 ! sum of !a + 139 = 180," a = 41.

a + b = 66 ext. ! of !41 + b = 66," b = 25.


Elementary students are not usually asked to record reasons in the manner done in theseexamples. But in the New Elementary Mathematics textbook, middle school students are ex-pected to present solutions exactly as above — middle school students use the same format aselementary school teachers!

This Teacher’s Solution format is used by

• Teachers in elementary school,

• Teachers and students in middle school.

Unknown angle problems in grades 5-7 also help introduce algebra. These geometry prob-lems expose students to the idea of using letters to stand for numbers (notice that the wholealphabet, not just x and y, are used!). They often require solving simple linear equations, asyou see in Example 1.4. This is a sneaky trick: giving students practice in algebra as they dogeometry. Solving for unknown angles also gives students a visual way to understand what itmeans to “solve for x” and appreciate why one would want to.

Homework Set 10

1. (Study the Textbook!) Give one-sentence answers to Ex-ercises 1.1, 1.2, and 1.3 in this section.

2. (Study the Textbook!) Turn to page 87 of Primary Math5A.• Do all of the problems on pages 87 and 88. Write

your answers as a single list of numbers, separatedby commas, but not labeled by the problem numbers.

• Note the variation: on pages 87 and 88, how manydi"erent letters are used to stand for numbers?

3. On page 109 of Primary Math 6B, answer Problems 35,36, and 37.

4. Answer Problem 2 on page 264 of NEM1. Show yourreasoning.

5. For Problem 3 on page 264 of NEM 1, write Teacher’sSolutions to parts b, d, e, g, and j.

6. For Problem 4 on the next page (page 265) write Teacher’sSolutions to parts b, c, f and g.

7. (Study the Textbook!) Compare the grade 6 and grade 7problems you did in this HW Set. Name two new fea-tures required in solving the grade 7 problems that are notpresent in the grade 6 problems.

geometric figures - singapore math · · 2014-06-0928 • geometric figures 1. ... two...

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